What is the value of k? (1) x^2 - px + q = (x - k) (x - r)x2−px+q=(x−k

Question

What is the value of k?

(1)  x^2 - px + q = (x - k) (x - r) 
(2)  r - p = 5

ryanstarr · Answer

This is a good example of an abstract quadratics problem, the type that tends to show up about once per exam if you're doing pretty well. It trades on your comfort with the algebra behind quadratics questions.

From a DS perspective, we can narrow the field of answers down to (C) and (E) pretty quickly. It's asking for the (numerical) value of  k , and while statement (1) mentions  k , it provides no numbers whatsoever. We could, if we multiplied out the right side, get relationships among  k ,  r ,  p , and  q ; however, I'd reserve such math for later pending a first look at statement (2), as we might be able to avoid it entirely. In any case,  statement (1) alone is insufficient.

Unlike statement (1), statement (2) does provide a concrete number; however, it provides no information about  k  whatsoever. So  statement (2) alone is insufficient  as well.

This brings us, as promised, to (C) vs. (E). Because we know from statement (1) that we can find an algebraic relationships among  k ,  r ,  p , and  q , and because statement (2) puts a number to the relationship between  r  and  p , it's a decent idea to FOIL out the right side of the equation in statement (1) and specifically examine the relationship between  k ,  r , and  p .
   x^2 - px + q = (x - k)(x - r) 
 x^2 - px + q = x^2 - rx - kx + kr 
 x^2 - px + q = x^2 - (r + k)x + kr

Because of the parallels between the two sides of the equations (and assuming that  k ,  r ,  p , and  q  are constants, which the GMAT will usually mention), we can infer that  r + k = p  and in turn that  r - p = -k . Since  r - p = 5  as well, we can say that  5 = -k  and that  -5 = k . Thus, both statements combined are sufficient, and  the answer is (C).

ktzsikka · Answer

Here is the OE  
 
 Solution:

Step 1: Analyse Question Stem 
We need to find the value of k.
No further information is given so let’s move to analysing the statements

Step 2: Analyse Statements Independently (And eliminate options) – AD/BCE 
 Statement 1 :  x^2 - px + q = (x - k) (x - r) 
• According to this statement:  x^2 – px + q = x^2 – (k+r) x + kr 
• Comparing both sides of the above equations, we get,
•  k+r = p ⟹ k = p - r 
• and,  kr = q ⟹ k= \frac{q}{r}  
• However, we don’t know the value of p, r and q. Therefore, we cannot determine the value of k from this statement alone.
Hence, statement 1 is NOT sufficient and we can eliminate answer Options A and D.
 Statement 2 :  r - p = 5 
• This statement doesn’t give any information about k at all. So, we will not be able to find k, using this statement alone.
Hence, statement 2 is also NOT sufficient and we can eliminate answer Option B.
Step 3: Analyse Statements by combining.
• From statement 1: 
•  k = p-r  
• And,  k = \frac{q}{r} 
• From statement 2: 
•  r -p = 5 ⟹ p – r = -5  
• On combining both statements, we get,
•  k = p -r = -5 
Thus, the correct answer is  Option C.  ( BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.)
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What is the value of k? (1) x^2 - px + q = (x - k) (x - r)x2−px+q=(x−k

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What is the value of k? (1) x^2 - px + q = (x - k) (x - r)x2−px+q=(x−k

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